Efficient and practical methods of simulating multivariate and multidimensional processes with specified cross‐spectral density are presented. When the cross‐spectral density matrix of ann‐variate process is specified, its component processes can be simulated as the sum of cosine functions with random frequencies and random phase angles. Typical examples of this type are the simulation, for the purpose of shaker test, of a multivariate process representing six components of the acceleration (due to, for example, a booster engine cutoff) measured at the base of a spacecraft and the simulation of horizontal and vertical components of earthquake acceleration. A homogeneous multidimensional process can also be simulated in terms of the sum of cosine functions with random frequencies and random phase angles. Examples of multidimensional processes considered here include the horizontal componentf0(t,x) of the wind velocity perpendicular to the axis (xaxis) of a slender structure, the vertical gust velocity fieldf0(x,y) frozen in space, and the boundary‐layer pressure fieldf0(x,y,t). Also, a convenient use of the present method of simulation in a class of nonlinear structural vibration analysis is described with a numerical example.